
'''simple docstring'''






from __future__ import annotations

from scipy.special import comb  # type: ignore

class 			UpperCAmelCase__    :



    def __init__(						self   :							Optional[Any],__A   :							list[tuple[float, float]]  ):
          _lowerCamelCase					: Tuple		    = list_of_points
          # Degree determines the flexibility of the curve.
          # Degree = 1 will produce a straight line.
          _lowerCamelCase					: int		    = len(__A  ) - 1



    def 	lowerCamelCase_	(						self   :							Optional[int],__A   :							float  ):
          assert 0 <= t <= 1, "Time t must be between 0 and 1."
          _lowerCamelCase					: list[float]		    = []
          for i in range(len(self.list_of_points  )  ):
                # basis function for each i
                output_values.append(
                    comb(self.degree,__A  ) * ((1 - t) ** (self.degree - i)) * (t**i)  )
          # the basis must sum up to 1 for it to produce a valid Bezier curve.
          assert round(sum(__A  ),5  ) == 1
          return output_values



    def 	lowerCamelCase_	(						self   :							int,__A   :							float  ):
          assert 0 <= t <= 1, "Time t must be between 0 and 1."

          _lowerCamelCase					: List[Any]		    = self.basis_function(__A  )
          _lowerCamelCase					: str		    = 0.0
          _lowerCamelCase					: str		    = 0.0
          for i in range(len(self.list_of_points  )  ):
                # For all points, sum up the product of i-th basis function and i-th point.
                x += basis_function[i] * self.list_of_points[i][0]
                y += basis_function[i] * self.list_of_points[i][1]
          return (x, y)



    def 	lowerCamelCase_	(						self   :							Optional[Any],__A   :							float = 0.01  ):
          from matplotlib import pyplot as plt  # type: ignore

          _lowerCamelCase					: list[float]		    = []  # x coordinates of points to plot
          _lowerCamelCase					: list[float]		    = []  # y coordinates of points to plot

          _lowerCamelCase					: Tuple		    = 0.0
          while t <= 1:
                _lowerCamelCase					: str		    = self.bezier_curve_function(__A  )
                to_plot_x.append(value[0]  )
                to_plot_y.append(value[1]  )
                t += step_size

          _lowerCamelCase					: List[str]		    = [i[0] for i in self.list_of_points]
          _lowerCamelCase					: Union[str, Any]		    = [i[1] for i in self.list_of_points]

          plt.plot(
              __A,__A,color="blue",label="Curve of Degree " + str(self.degree  ),)
          plt.scatter(__A,__A,color="red",label="Control Points"  )
          plt.legend()
          plt.show()


if __name__ == "__main__":
 import doctest

 doctest.testmod()

 BezierCurve([(1, 2), (3, 5)]).plot_curve()  # degree 1
 BezierCurve([(0, 0), (5, 5), (5, 0)]).plot_curve()  # degree 2
 BezierCurve([(0, 0), (5, 5), (5, 0), (2.5, -2.5)]).plot_curve()  # degree 3





"""simple docstring"""




from __future__ import annotations

from decimal import Decimal

from numpy import array



def        a__					(							snake_case__							)		->   list[list[float]]:
			lowerCamelCase      =   Decimal

			# Check if the provided matrix has 2 rows and 2 columns
			# since this implementation only works for 2x2 matrices
			if len(snake_case__							) == 2 and len(matrix[0]							) == 2 and len(matrix[1]							) == 2:
						# Calculate the determinant of the matrix
						lowerCamelCase      =   float(
						    d(matrix[0][0]							) * d(matrix[1][1]							) - d(matrix[1][0]							) * d(matrix[0][1]							)							)
						if determinant == 0:
									raise ValueError("""This matrix has no inverse."""							)

						# Creates a copy of the matrix with swapped positions of the elements
						lowerCamelCase      =   [[0.0, 0.0], [0.0, 0.0]]
						lowerCamelCase ,  lowerCamelCase      =   matrix[1][1], matrix[0][0]
						lowerCamelCase ,  lowerCamelCase      =   -matrix[1][0], -matrix[0][1]

						# Calculate the inverse of the matrix
						return [
						    [(float(d(snake_case__							)							) / determinant) or 0.0 for n in row] for row in swapped_matrix
						]
			elif (
			    len(snake_case__							) == 3
			    and len(matrix[0]							) == 3
			    and len(matrix[1]							) == 3
			    and len(matrix[2]							) == 3
			):
						# Calculate the determinant of the matrix using Sarrus rule
						lowerCamelCase      =   float(
						    (
						        (d(matrix[0][0]							) * d(matrix[1][1]							) * d(matrix[2][2]							))
						        + (d(matrix[0][1]							) * d(matrix[1][2]							) * d(matrix[2][0]							))
						        + (d(matrix[0][2]							) * d(matrix[1][0]							) * d(matrix[2][1]							))
						    )
						    - (
						        (d(matrix[0][2]							) * d(matrix[1][1]							) * d(matrix[2][0]							))
						        + (d(matrix[0][1]							) * d(matrix[1][0]							) * d(matrix[2][2]							))
						        + (d(matrix[0][0]							) * d(matrix[1][2]							) * d(matrix[2][1]							))
						    )							)
						if determinant == 0:
									raise ValueError("""This matrix has no inverse."""							)

						# Creating cofactor matrix
						lowerCamelCase      =   [
						    [d(0.0							), d(0.0							), d(0.0							)],
						    [d(0.0							), d(0.0							), d(0.0							)],
						    [d(0.0							), d(0.0							), d(0.0							)],
						]
						lowerCamelCase      =   (d(matrix[1][1]							) * d(matrix[2][2]							)) - (
						    d(matrix[1][2]							) * d(matrix[2][1]							)
						)
						lowerCamelCase      =   -(
						    (d(matrix[1][0]							) * d(matrix[2][2]							)) - (d(matrix[1][2]							) * d(matrix[2][0]							))
						)
						lowerCamelCase      =   (d(matrix[1][0]							) * d(matrix[2][1]							)) - (
						    d(matrix[1][1]							) * d(matrix[2][0]							)
						)
						lowerCamelCase      =   -(
						    (d(matrix[0][1]							) * d(matrix[2][2]							)) - (d(matrix[0][2]							) * d(matrix[2][1]							))
						)
						lowerCamelCase      =   (d(matrix[0][0]							) * d(matrix[2][2]							)) - (
						    d(matrix[0][2]							) * d(matrix[2][0]							)
						)
						lowerCamelCase      =   -(
						    (d(matrix[0][0]							) * d(matrix[2][1]							)) - (d(matrix[0][1]							) * d(matrix[2][0]							))
						)
						lowerCamelCase      =   (d(matrix[0][1]							) * d(matrix[1][2]							)) - (
						    d(matrix[0][2]							) * d(matrix[1][1]							)
						)
						lowerCamelCase      =   -(
						    (d(matrix[0][0]							) * d(matrix[1][2]							)) - (d(matrix[0][2]							) * d(matrix[1][0]							))
						)
						lowerCamelCase      =   (d(matrix[0][0]							) * d(matrix[1][1]							)) - (
						    d(matrix[0][1]							) * d(matrix[1][0]							)
						)

						# Transpose the cofactor matrix (Adjoint matrix)
						lowerCamelCase      =   array(snake_case__							)
						for i in range(3							):
									for j in range(3							):
												lowerCamelCase      =   cofactor_matrix[j][i]

        # Inverse of the matrix using the formula (1/determinant) * adjoint matrix
						lowerCamelCase      =   array(snake_case__							)
						for i in range(3							):
									for j in range(3							):
												inverse_matrix[i][j] /= d(snake_case__							)

        # Calculate the inverse of the matrix
						return [[float(d(snake_case__							)							) or 0.0 for n in row] for row in inverse_matrix]
			raise ValueError("""Please provide a matrix of size 2x2 or 3x3."""							)



